The problems of predicting heat and humidity conditions in various mine structures, as well as taking into account induced pollution of frozen soil and pollutant migration in permafrost strata need new mathematical models. It is required that these models take into account to a fuller extent features of physical freezing and thawing processes in pore water. It is also of the current concern to develop new software tools and methods for the numerical solution of model problems on heat–moisture–salt transfer in multi-dimensional domains with regard to phase ice–solution transition. This article proposes a method to solve the problems of heat and and mass transfer with phase transformation (freezing–thawing). The method involves splitting of the initial problem of heat and mass transfer with phase transformation with respect to physical processes. Such processes are assumed to be diffusion transfer of heat, moisture and soluble salt between points of meshing, as well as the processes of redistribution in the point neighborhoods which are assumed to be isolated systems. The developed algorithm allows calculating complex processes including phase transformations while integrating solutions of nonlinear equations of thermal conduction in application packages. Irrespective of space parameters (at the stage of phase transformations), the algorithm makes it possible to use equations of phase equilibrium both for the processes with narrow and wide temperature range of the phase transfer zone (Stefan problem type), including multi-front and multidimensional processes, in the framework of the common methodical approach. Tests of the similarity and numerical solutions show a satisfactory agreement (accurate to 2%). This offers good prospects for further applications.


Heat and mass transfer, phase transformations, splitting with respect to physical processes, solution method.

Issue number: 12
Year: 2018
UDK: 622.45:536.421
DOI: 10.25018/0236-1493-2018-12-0-57-64
Authors: Popov V. I., Kurilko A. S.

About authors: Popov V.I., Candidate of Technical Sciences, Senior Researcher, e-mail:, Kurilko A.S., Doctor of Technical Sciences, Professor, Deputy Director for Science, Chersky Mining Institute of the North, Siberian Branch, Russian Academy of Sciences, 677000, Yakutsk, Russia.


1. Budak B. M., Vasil'ev F. P. Egorova A. T. Ob odnom variante neyavnoy raznostnoy skhemy s lovley fronta v uzel setki dlya resheniya zadach tipa Stefana [A variant of implicit difference scheme with front capturing at mesh point for solving Stefan-type problems], Vychislitel'nye metody i programmirovanie. Issue 4], Moscow, Izd-vo MGU, 1967, pp. 231—241.

2. Budak B. M., Gol'dman N. L., Uspenskiy A. B. Raznostnye skhemy s vypryamleniem frontov dlya resheniya mnogofrontovykh zadach tipa Stefana [Difference schemes with strengthened fronts for solving multi-front Stefan problems], Doklady AN SSSR. 1966. Vol. 167, no 4, pp. 735—738. [In Russ].

3. Kolesnikov A. G. K izmeneniyu matematicheskoy formulirovki zadachi o promerzanii grunta [Changing mathematical formulation of problems on soil freezing], Doklady AN SSSR. 1952. Vol. 82, no 6, pp. 889—892. [In Russ].

4. Lykov A. V. Yavleniya perenosa v kapillyarno-poristykh telakh [Transfer phenomena in capillary–porous bodies], Moscow, Izd-vo tekhn.-teoret. lit-ry, 1954.

5. Permyakov P. P., Romanov P. G. Teplo- i soleperenos v merzlykh nenasyshchennykh gruntakh [Heat and salt transfer in unsaturated frozen soil], Yakutsk, Izd-vo YAF SO RAN, 2000.

6. Taylor G. S., Luthin J. N. A model for coupled heat moisture transfer during soil freezing. Canad. Geotechnical J. 1978. V. 15. P. 548—555.

7. Jame Y. W., Norum D. J. Heat and mass transfer in freezing unsaturated porous medium. Water Resour. Rec. 1980. V. 16. No 4. P. 811—819.

8. Samarskiy A. A., Moiseenko B. D. Ekonomichnaya skhema skvoznogo scheta dlya mnogomernoy zadachi Stefana [Economic end-to-end calculation scheme for multidimensional Stefan problem]. Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki. 1965. vol. 5, no 5, pp. 816—827. [In Russ].

9. Komarov I. A. Termodinamika i teplomassoobmen v dispersnykh merzlykh porodakh [Thermodynamics and heat and mass exchange in dispersed frozen rocks], Moscow, Nauchnyy mir, 2003.

10. Marchuk G. I. Metody vychislitel'noy matematiki [Methods of computational mathematics], Moscow, Nauka, 1977.

11. Karpov I. K. Fiziko-khimicheskoe modelirovanie na EVM v geokhimii [Computer-aided physicochemical modeling in geochemistry], Novosibirsk, Nauka, 1981.

12. Hsiao J. S. An efficient algoritm for finite-difference analyses of heat transfer with melting and solidification. Numer. Heat Transfer. 1985. V.8, No 6. P. 653—666.

13. Khokholov Yu. A., Solov'ev D. E. Matematicheskoe modelirovanie teplovykh protsessov v gornykh vyrabotkakh shakht i rudnikov Severa [Mathematical modeling of thermal processes in mines in the North], Novosibirsk, Akademicheskoe izd-vo «Geo», 2013, 185 p.

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